For clarity, we start with assuming a known 3D view-to-view warp \(W_{t_1\rightarrow t_2}\) , outlined in the last sub-section.
The 2D correspondence \(\mathbf {u}_{t_2}\)  given \(\mathbf {u}_{t_1}\)  can be obtained by three steps, which we describe as “warp-integrate-project”.
In the “warp” step, given the pixel location \(\mathbf {u}_{t_1}\)  and camera \(\pi _{t_1}\) , we sample points on the ray passing from the camera center through the pixel \(\pi _{t_1}^{-1}(\mathbf {u}_{t_1})\) . Then, we warp the sampled points toward their 3D correspondences in the target frame using the known 3D warp \(W_{t_1\rightarrow t_2}\) .
In the “integrate” step, we compute the expected 3D location for the source samples weighted by the probability mass \(w_{t_1}\)  by volume rendering, as per NeRF [1]}.
We can use densities from either source or target frame, a choice that we find insensitive in practice.
In our formulation, we use the densities from the source frame.
Finally, in the “project” step, we project the expected 3D location to the target frame through the target camera \(\pi _{t_2}\) .
The “warp-integrate-project” process can be written as
\(\mathbf {u}_{t_2}=\pi _{t_2}\bigg (\mathop {\mathbb {E}}_{\mathbf {x}_{t_1} \in \pi _{t_1}^{-1}(\mathbf {u}_{t_1})}\big [w_{t_1}(\mathbf {x}_{t_1}) \cdot W_{t_1 \rightarrow t_2}(\mathbf {x}_{t_1})\big ]\bigg ).\) 
